To demonstrate the impact of our disentangled framework and local fairness we use a synthetic dataset, which follows [1]}.
Samples \((x_i, y_i, z_i)^n_{i=1}\)  are generated as follows: For each sample \(i\) , we first generated two random seeds, \(r_i, yo_i \in \lbrace 0, 1\rbrace \) .
We then sample \(v_i\sim \mathcal {N}(r_i,1)\)  and \(u_{r_1}, \dots , u_{r_d}\sim \mathcal {N}(v_i,1)\)  (dependently on \(v_i\) ).
We also sample \(w_i\sim \mathcal {N}(v_i, 1)\)  dependently on \(v_i\) . Next we sample
\(vo_i\sim \mathcal {N}(yo_i,1)\)  and \(u_{yo_1}, \dots , u_{yo_d}\sim \mathcal {N}(vo_i,1)\)  dependently on \(vo_i\) .
Each sample \(i\)  is then defined as follows: \(x^{(i)} = (r_i,u_{r_1}, \dots , u_{r_d}, u_{yo_1}, \dots , u_{yo_d})\) , \(y_r^{(i)} =\mathbb {1}[w_i > 0]\) , \(y_o^{(i)} = yo_i\)  and \(z^{(i)} = r\)  (where \(\mathbb {1}\)  denotes an indicator function).
To construct the final dataset, we define \(p_z\)  to be the portion of biased samples in the dataset and choose the final labels \(Y\)  such that \(p_y\)  portion of the \(n\)  samples comes from the biased distribution \(y_r^{(i)}\)  and the rest from \(y_o^{(i)}\) .
For this experiment, we set \(p_z = 0.5\)  for train data and \(p_z = 0\)  for test data. This is done to demonstrate a case where the training phase is exposed to biased data (\(50\%\)  of samples) and our goal is to learn a fair predictor. In our setting, \(d=25\) . We use 1000 (resp. 500) train (resp. test) samples.
